The rotary engine has at least one casing cavity (10) of non-round cross section and one rotor (12) fixed to the drive shaft 13. It rotates with this shaft but is able to move transversely on it in a combined rotary and translational motion. The non-round cross section of the casing cavity is such that as the rotor moves in this combined motion, it divides the casing cavity into two working cavities (A,B) of increasing and decreasing volumes. Part of this cross section may be semicircular, with the turning point of the drive shaft off-centre on the diameter of the semicircle.
